The subject says it all. I would like to know if Proposition 3.1 in
Arthur-Clozel's book on the trace formula holds for local fields of positive
characteristic.

Thanks!

EDIT: Here is Prop 3.1 of Arthur-Clozel: (Notation will be explained after the statement)

Proposition 3.1

Assume $\phi\in C_c^\infty(GL_n(E))$. Then there exists $f\in C_c^\infty(GL_n(F))$ such
that,
for regular $\gamma\in GL_n(F)$, $O_\gamma(f) = 0$ if $\gamma$ is not a norm and
$O_\gamma(f) = TO_{\sigma\delta}(\phi)$ if $\gamma = N\delta$.

Notation

$E/F$ is an finite unramified extension of local fields (hence cyclic)
of degree $r$. Let $\sigma\in Gal(E/F)$ be a generator.

Then there is a norm map
$N : \sigma\text{-conjugacy classes in } G(E)\to \text{conjugacy classes in }
G(F)$, where we say that $\delta, \delta'\in G(E)$ are $\sigma$-conjugate if there is $h\in G(E)$ such that $\delta' = h^{-1}\delta\sigma(h)$. The map $N$ is defined by sending the class of
$\delta$ to the class of $\delta\sigma(\delta)\cdots\sigma^{r-1}\delta$.

$O_\gamma(f)$ is an orbital integral: $O_\gamma(f) = \int_{GL_n(F)_\gamma\backslash GL_n(F)}f(g^{-1}\gamma g)\ dg$, where $GL_n(F)_\gamma\subset GL_n(F)$ is the centralizer of $\gamma$.